What is the physical meaning of a mass times a distance squared? Why is such a product of interest in the applications?
For straight-line motion, the mass of a particle is the constant of proportionality between the applied force F, and the resulting acceleration a, a relationship captured in Newton's second law .
This section will show that is the constant of proportionality between an applied torque, and a resulting angular acceleration. (A force F applied at a distance from a pivot point generates a torque (or twist) defined as , where R is a vector from the pivot point to the point where the force is applied.)
Let an element of mass rotate about the -axis at a distance from the axis. Measure the angle through which the element has turned by . Then the quantities in Table 6.6.2 are relevant.
Angular position
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Angular velocity
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Angular acceleration
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Arc length:
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Rim (or linear) speed:
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Linear acceleration on rim:
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Table 6.6.2 Rotation of about -axis
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If and , then for a force F tangential to the circle through which rotates,
= = = =
Multiply the scalar form of Newton's second law by to obtain
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The constant of proportionality between , the magnitude of the torque, and the resulting angular acceleration is the scalar . This quantity measures the "rotational inertia" and is the content of the integrals that define and , the moments of inertia about the - and -axes, respectively.